Showing papers on "Random vibration published in 2000"

Optimal nonlinear stochastic control of hysteretic systems

Abstract: A strategy for optimal nonlinear stochastic control of hysteretic systems with parametrically and/or externally random excitations is proposed and illustrated with an example of a single-degree-of-freedom system. A hysteretic system subject to random excitation is first replaced by a nonlinear nonhysteretic stochastic system, from which an Ito equation for the averaged total energy of the system as a 1D controlled diffusion process is derived by using the stochastic averaging method of energy envelope. For a given performance index, a Hamilton-Jacobi-Bellman equation is then established based on the stochastic dynamical programming principle and solved to yield the optimal control force. Finally, the statistics of the responses of uncontrolled and controlled systems and those of the optimal control force are predicted analytically. Comparison with the modified optimal polynomial controller indicates that the proposed strategy is more effective and efficient.

Durability/reliability of BGA solder joints under vibration environment

Abstract: The objective of the present study is to develop an experimentally validated vibration fatigue damage model for a plastic ball grid array (PBGA) solder joint assembly A 3-D modeling technique is used to simulate the vibration responses of the BGA packages soldered onto a printed wiring assembly (PWA) In the study, a method to determine the stresses/strains of BGA solder joints resulting from exposure of the PWA to random vibration environments is first described Linear static and dynamic finite element analyses using MSC/NASTRAN/sup TM/ computer code, combined with a volume-weighted average technique, are then conducted to calculate the effective strains of the solder joints In the calculation process, several in-house developed Fortran programs, in conjunction with the outputs obtained from MSC/NASTRAN/sup TM/ static and frequency response analyses, are used to perform the required computations A vibration fatigue life model, evolved from an empirically derived formula of universal slopes based on high-cycle fatigue test data, is established This model combined with a three-band technique and the derived solder effective strain is then used to predict the PBGA solder joint survivability/durability This prediction is compared to in-house test results to qualitatively calibrate (in the sense of conservation) the proposed PBGA solder joint vibration fatigue damage model This validated model is then recommended to serve as an effective tool to determine the integrity of the PBGA solder joints during vibration Selecting more study cases with various package sizes, solder ball configurations, vibration profiles to further calibrate this vibration fatigue damage model is also recommended An example of a 313-pin PBGA is illustrated in the present study

Comparison of Nonlinear Random Response Using Equivalent Linearization and Numerical Simulation

Abstract: A recently developed finite-element-based equivalent linearization approach for the analysis of random vibrations of geometrically nonlinear multiple degree-of-freedom structures is validated. The validation is based on comparisons with results from a finite element based numerical simulation analysis using a numerical integration technique in physical coordinates. In particular, results for the case of a clamped-clamped beam are considered for an extensive load range to establish the limits of validity of the equivalent linearization approach.

Bounded parametric control of random vibrations

Abstract: The dynamic programming approach is used to study a feedback control problem for a randomly excited single–degree–of–freedom system. The available actuator for control can provide temporal stiffness variations for the system, which are of a bounded magnitude. An analytical solution is obtained for the corresponding Hamilton–Jacobi–Bellman equation for the expected response energy, which should be minimized according to the integral cost criterion. While this solution is valid within some parts of the phase plane only, it extends to the whole phase plane with increasing time–interval of control. Thus the exact explicit expression for the optimal control law is obtained for the case where steady–state response is to be controlled. This law requires feedback–controlled switching stiffness between the given bounds. Stationary random vibration of the system with this control law is studied then, by a direct energy balance approach and by a stochastic averaging method. The latter of these provides a more extensive description of the response, which may provide reliability estimates for the controlled system, but this asymptotic approach is valid for the limiting case of a weak control only. The direct energy balance predicts only the expected response energy, or the mean square displacement. However, its range of applicability is expected to be less restrictive with respect to the maximal magnitude of the system9s stiffness variations: the non–dimensional variations may not be much smaller than unity. The analytical studies are extended to the case of a controlled system with a nonlinear restoring force.

Explorations of the Phase-Space Linearization Method for Deterministic and Stochastic Nonlinear Dynamical Systems

Abstract: A new numeric-analytic phase-space linearization (PSL) schemefor a class of nonlinear oscillators with continuous vector fields isinvestigated in this study. The essence of the PSL method is to replacethe nonlinear vector field by a set of conditionally linear ones, eachvalid either over a short segment of the evolving trajectory or(equivalently) over a sufficiently small interval of time. This conceptmay be usefully exploited to arrive at certain explicit and implicitintegration schemes for analyses and simulations. The explicit schemes,which are found to have ready extensions to systems under stochasticinputs, are first numerically implemented for a few oft-used nonlineardynamical systems under (deterministic) sinusoidal inputs. An estimateof an upper bound to the local error in terms of the chosen time stepsize is provided. The explicit scheme of local linearization is nextextended to nonlinear oscillators under stochastic excitations, namelywhite noise processes, which are formal derivatives of one or acombination of Gauss–Markov processes. Since the PSL approach is todecompose the nonlinear operator into a set of linear operators, theprinciples of linear random vibration may be suitably exploited toarrive at a faster Monte-Carlo scheme for computing the responsestatistics, both in stationary and nonstationary regimes. A fewexamples, based on Ueda's and Duffing–Holmes' oscillators, arepresented and compared with exact solutions, whenever available, toverify the correctness and versatility of the proposed schemes.

Shock and thermal cycling synergism effects on reliability of CBGA assemblies

Abstract: BGAs are now packages of choice especially for higher I/O counts for commercial applications and are also being considered for use in military and aerospace. Thermal cycling characteristics of BGA assemblies have been widely reported. Thermal cycling represents the on-off environmental condition for most electronic products and therefore is a key factor that defines reliability. As a result, much data is available for accelerated thermal cycle conditions, but very limited data is available on vibration and shock representative of aerospace applications. Test vehicles with daisy chain plastic and ceramic BGAs (CBGAs) ranging from 256 to 625 I/O count were subjected to random vibration/shock representative of a spacecraft launch environment. The effect of board rigidity on behavior was also investigated by adding strips to or bonding of board to an aluminum plate. This paper compares accelerated thermal cycles-to-failure data under four temperature ranges before and after thermal random vibration for CBGAs with 361 and 625 I/Os. Stress and strain projections by finite element analysis are also presented.

Triply Coupled Vibration of Asymmetric Wall-Frame Structures

Abstract: This paper presents an analysis of the triply coupled vibration of asymmetric wall-frame structure in tall buildings. The governing equation of the natural vibration and its corresponding eigenvalue problem, which is a set of equations for flexural-shear vibrations in laterally orthogonal directions coupled with a warping St. Venant torsional vibration, are developed. By applying the Galerkin method, a generalized approximate approach is developed for the analysis of coupled vibration and for determining the natural frequencies and associated mode shapes of the structure in triply coupled vibration. The results of the proposed method for the example structure show good agreement with those of the FEM analysis. The proposed method has been shown to provide a simple and efficient, yet accurate, means of coupled vibration analysis of asymmetric tall building structures.

Random vibration studies of an SDOF system with shape memory restoring force

Abstract: Intelligent and adaptive material systems and structures have become very important in engineering applications. The basic characteristic of these systems is the ability to adapt to the environmental conditions. A new class of materials with promising applications in structural and mechanical systems is shape memory alloy (SMA). The mechanical behavior of shape memory alloys in particular shows a strong dependence on temperature. This property provides opportunities for the utilization of SMAs in actuators or energy dissipation devices. However, the behavior of systems containing shape memory components under random excitation has not yet been addressed in the literature. Such a study is important to verify the feasibility of using SMAs in structural systems. In this work a nondeterministic study of the dynamic behavior of a single-degree-of-freedom (SDOF) mechanical system, having a Nitinol spring as a restoring force element is presented. The SMA spring is characterized using a one-dimensional phenomenological constitutive model based on the classical Devonshire theory. Response statistics for zero mean random vibration of the SDOF under a wide range of temperature is obtained. Furthermore, nonzero mean analysis of these systems is carried out.

Flow induced by a randomly vibrating boundary

Abstract: We study the flow induced by random vibration of a solid boundary in an otherwise quiescent fluid. The analysis is motivated by experiments conducted under the low level and random effective acceleration field that is typical of a microgravity environment. When the boundary is planar and is being vibrated along its own plane, the variance of the velocity field decays as a power law of distance away from the boundary. If a low frequency cut-off is introduced in the power spectrum of the boundary velocity, the variance decays exponentially for distances larger than a Stokes layer thickness based on the cut-off frequency. Vibration of a gently curved boundary results in steady streaming in the ensemble average of the tangential velocity. Its amplitude diverges logarithmically with distance away from the boundary, but asymptotes to a constant value instead if a low frequency cut-off is considered. This steady component of the velocity is shown to depend logarithmically on the cut-off frequency. Finally, we consider the case of a periodically modulated solid boundary that is being randomly vibrated. We find steady streaming in the ensemble average of the first order velocity, with flow extending up to a characteristic distance of the order of the boundary wavelength. The structure of the flow in the vicinity of the boundary depends strongly on the correlation time of the boundary velocity.

Whole-spacecraft vibration isolation for broadband attenuation

Abstract: Launch vehicles impart high levels of vibration to spacecraft during launch. The vibration environments are defined over several frequency bands: (1) transient vibration <80 Hz, (2) random vibration 20 to 2000 Hz, and (3) pyrotechnic shock 100 to 10000 Hz. Loads from transient vibration define spacecraft design of primary structures such as spacecraft bus, solar panel and antenna supports, instrument mounts, etc. Loads from random vibration define the design for spacecraft light structures such as antennas and solar panels, and shock loads define the design of electronic components and instruments. The spacecraft must survive the combination of all vibration environments. This requires spacecraft structures, instruments, and components to be designed to minimize vibration across a broad frequency range. Spacecraft are designed for the short launch to orbit, which is well beyond the requirements for on-orbit performance. A better choice is to reduce the magnitude of the high launch loads across all frequency bands and design smaller and less costly spacecraft. SoftRide systems are under development for the first and second OrbitaVSuborbital Program (OSP) launches and for the TaurusNTI launch. Additionally, isolation systems are being designed for larger liquid-fueled launch vehicles. This isolation system technology will greatly further the goal of better, faster, cheaper, and lighter spacecraft.

Joint and Cross Acceptances for Cross-Flow-Induced Vibration—Part I: Theoretical and Finite Element Formulations

Abstract: The theoretical development of the acceptance integral method to estimate the random vibration of structures subject to turbulent flow is critically reviewed and put onto a firm mathematical basis. Closed-form solutions for the joint acceptances for cross-flow-induced vibration of one-dimensional structures are derived for two special cases of spring-supported and simply supported beams. These are used to check results from a finite element formulation of the acceptance integrals for one-dimensional structures with arbitrary boundary conditions, and for arbitrary correlation lengths. Agreements between the finite element and closed-form solutions are excellent.

Optimal Design of Uncertain Systems Under Stochastic Excitation

Abstract: The structural synthesis problem of uncertain linear systems subjected to uncertain loads is considered. Uncertain system parameters are modeled as random variables with a prescribed joint probability density function, whereas the loads are modeled as stochastic processes. Second-order probabilistic descriptors are combined with approximate extreme response theories to obtain reliability estimates for the systems. A multicriterion optimal design methodology, based on a preference aggregation rule, is used in this formulation. Optimization is carried out by generating and solving a sequence of explicit approximate problems. The uncertainty in system parameters is taken explicitly in the analysis, and its effect is investigated on the optimal design. It is shown that uncertainties are important because they can change the system reliability significantly. Therefore, in these cases, the effect of uncertainty in the model parameters must be considered explicitly during the design process. An example problem is presented to illustrate the performance of the proposed methodology.

A new stationary PDF approximation for non-linear oscillators

Abstract: A new method is presented for approximating the stationary probability density function of the response of a general class of non-linear single-degree-of-freedom dynamical systems subjected to additive stochastic white noise excitation. The method is based on finding the best probability density function (PDF) from a parameterized class of trial non-Gaussian PDFs by minimizing a weighted norm of the Fokker–Planck–Kolmogorov equation error. The proposed procedure yields simple expressions in terms of one-dimensional integrals for determining desired probabilistic characteristics of the system response, such as moments and mean outcrossing rates. Examples illustrating the applicability and accuracy of the method include a system modeling the rolling motion of a ship and a Duffing oscillator with non-linear damping. Comparisons are made with some other approximate methods, including equivalent linearization, partial linearization, equivalent non-linearization, and dissipation energy balancing methods, that show that the new method yields substantially improved estimates for the expected outcrossing rates of the response. These outcrossing rates are crucial for evaluating the reliability of the system. In contrast, the equivalent non-linearization and the dissipation energy balancing methods, known to provide the most accurate estimates for the mean-square response, give very poor estimates of the mean outcrossing rates as the number of level outcrossings decreases.

Variations in steepness of the probability density function of beam random vibration

Abstract: Dynamic behaviour of a beam, subjected to stationary random excitation, has been investigated for the situation in which the response is different from the model of a Gaussian random process. The study was restricted to the case of symmetric non-Gaussian probability density functions of beam vibrations. There are two possible causes of deviations of the system response from the Gaussian model: the first, nonlinear behaviour, concerns the system itself and the second is external when the excitation is not Gaussian. Both cases have been considered in the paper. To clarity the conclusions for each case and to avoid interference of these different types of system behaviour, two beam structures, clamped-clamped and cantilevered, have been studied. A numerical procedure for prediction of the nonlinear random response of a clamped-clamped beam under the Gaussian excitations was based on a linear modal expansion. Monte Carlo simulation was undertaken using Runge–Kutta integration of the generalised coordinate equations. Probability density functions of the beam response were analysed and approximated making use of different theoretical models. An experimental study has been carried out for a linear system of a cantilevered beam with a point mass at the free end. A pseudo-random driving signal was generated digitally in the form of a Fourier expansion and fed to a shaker input. To generate a non-Gaussian excitation a special procedure of harmonic phase adjustment was implemented instead of the random choice. In so doing, the non-Gaussian kurtosis parameter of the beam response was controlled.

A tutorial on design analysis using von Mises stress in random vibration environments

Abstract: The von Mises stress is often used as the metric for evaluating design margins, particularly for structures made of ductile materials. While computing the von Mises stress distribution in a structural system due to a deterministic load condition may be straightforward, difficulties arise when considering random vibration environments. As a result, alternate methods are used in practice. One such method involves resolving the random vibration environment to an equivalent static load. This technique, however, is only appropriate for a very small class of problems and can easily be used incorrectly. Monte Carlo sampling of numerical realizations that reproduce the second order statistics of the input is another method used to address this problem. This technique proves computationally inefficient and provides no insight as to the character of the distribution of von Mises stress. This tutorial describes a new methodology to investigate the design reliability of structural systems in a random vibration environment. The method provides analytic expressions for root mean square (RMS) von Mises stress and for the probability distributions of von Mises stress which can be evaluated efficiently and with good numerical precision. Further, this new approach has the important advantage of providing the asymptotic properties of the probability distribution. A brief overview of the theoretical development of the methodology is presented, followed by detailed instructions on how to implement the technique on engineering applications. As an example, the method is applied to a complex finite element model of a Global Positioning Satellite (GPS) system. This tutorial presents an efficient and accurate methodology for correctly applying the von Mises stress criterion to complex computational models. The von Mises criterion is the traditional method for determination of structural reliability issues in industry.